We now show that \(S_H\)  does not contain both \(a_1^i\)  and \(A_1^i[1]\)  for any \(i\) , \(1 \le i \le k\) .
The same argument can be applied to other sets of row vertices \(A_2\) , \(B_1\) , and \(B_2\) .
The vertex \(a_{1}^{i}\)  in \(H^{2}_{x,y}\)  covers some bit gadget vertices and the vertex \(A_{1}^{i}[1]\)  covers the
same row vertices in \(A_{2}\)  as the vertex \(a_{1}^{i}\)  does in \(G_{x,y}\) . Note that \(a_{1}^{i}\)  does not cover any row vertex in \(H^{2}_{x,y}\) .
We know from [1]} that the MDS of \(G_{x,y}\)  has the property that the bit gadget vertices provide
coverage for all bit gadget vertices. This is a local argument and it also holds for the MDS of \(H^{2}_{x,y}\) 
since the subgraph induced by the bit gadget vertices in \(H^{2}_{x,y}\)  is the same as the subgraph induced in \(G_{x,y}\) .
Therefore, \(a_{1}^{i}\)  and \(A_1^i[1]\)  cannot belong to \(S_{H}\)  because if they do, we can remove
\(a_1^i\)  and still have a dominating set of \(H_{x, y}\) , contradicting the fact that \(S_H\)  is a minimum dominating set of \(H^2_{x, y}\) .
